Trajectory Position At A Given Time

I am writing a graphics program to model projectile motion of a ball being thrown. What I am looking for is an equation that will give a specific X and Y coordinate, in the parabolic path, for any specific time it is given, and a arbitratry velocity and angle.

For example, if I give it some arbitrary velocity and angle, I would like to be able to figure out how high in the air the ball is and how far it has traveled horizontally at 5 seconds, 13.4 seconds or any other random time.

What sort of simplifying assumptions are you making? A flat earth, constant acceleration of gravity w.r.t. altitude, no air drag? Since you mention "parabolic path," I imagine you are going with all of these idealizations.

If you idealize (simplify) things enough, you can make use of closed-form solutions that were worked out with calculus centuries ago. If you are going to make it more realistic, you will probably have to resort to numerical approximation methods.